Abstract : The Keller-Segel partial differential equation is a two-dimensional model for chemotaxis. When the total
mass of the
initial density is one, it is known to exhibit blow-up in finite time as soon as the sensitivity $\chi$
of bacteria to
the chemo-attractant is larger than $8\pi$. We investigate its approximation by a system of $N$
particles interacting through a singular attractive kernel in the drift term.
In the very subcritical case $\chi<2\pi$, the diffusion strongly dominates this singular drift:
we obtain existence
for the particle system and prove that its flow of empirical measures converges, as $N\to\infty$
and up to extraction of a subsequence, to a weak solution of the Keller-Segel equation.
We also show that for any $N\ge 2$ and any value of $\chi>0$, pairs of particles do collide with positive
probability: the singularity of the drift is indeed visited.
Nevertheless, when $\chi<2\pi N$, it is possible to control the drift and obtain existence of the particle system
until the first time when at least three particles collide. We check that this time is a.s. infinite,
so that global existence holds for the particle system,
if and only if $\chi\leq 8\pi(N-2)/(N-1)$.
Finally, we remark that in the system with $N=2$ particles, the difference between the two positions
provides a natural two-dimensional generalization of Bessel processes, which we study in details.